令人惊讶的简单光谱
Surprisingly Simple Spectra
摘要：
研究了用限制Schur多项式描述的${\cal N}=4$ super Yang-Mills理论中算子反常维数的大N极限.我们专注于由Young图标记的具有两列（均为长列）的运算符，因此这些运算符的经典维数为O（N）。在大N时，这两个列运算符彼此混合，但与具有$n\ne 2$列的运算符解耦。平面近似不捕获大N动力学。对于使用2、3或4种杂质构建的算子，明确评价扩张算子。在所有这三种情况下，在一定的限制下，伸缩算子是一个二阶导数的晶格版本，晶格从杨氏图本身出现。对单圈扩张算子进行数值对角化。所有特征值都是$8g_{YM}^2$的整数倍，并且在谱中存在有趣的简并。我们获得的频谱的一个循环反常维数运营商是由谐振子的集合再现。这种与谐振子的等价性推广了BPS扇区中已知的巨引力子结果，并进一步暗示了由单圈大$N$伸缩算子定义的哈密顿量是可积的。这是一个可积扩张算子的例子，通过对平面图和非平面图求和得到。
英文摘要：
The large N limit of the anomalous dimensions of operators in ${\cal N}=4$ super Yang-Mills theory described by restricted Schur polynomials, are studied. We focus on operators labeled by Young diagrams that have two columns (both long) so that the classical dimension of these operators is O(N). At large N these two column operators mix with each other but are decoupled from operators with $n\ne 2$ columns. The planar approximation does not capture the large N dynamics. For operators built with 2, 3 or 4 impurities the dilatation operator is explicitly evaluated. In all three cases, in a certain limit, the dilatation operator is a lattice version of a second derivative, with the lattice emerging from the Young diagram itself. The one loop dilatation operator is diagonalized numerically. All eigenvalues are an integer multiple of $8g_{YM}^2$ and there are interesting degeneracies in the spectrum. The spectrum we obtain for the one loop anomalous dimension operator is reproduced by a collection of harmonic oscillators. This equivalence to harmonic oscillators generalizes giant graviton results known for the BPS sector and further implies that the Hamiltonian defined by the one loop large $N$ dilatation operator is integrable. This is an example of an integrable dilatation operator, obtained by summing both planar and non-planar diagrams.
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